Elementary Mathematical Astronomy, Barlow & Bryan

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ELEMENTARY. MATHEMATICAL ASTRONOMY,

EXAMPLES AND EXAMINATION PAPERS.

C. W. 0. ^ABLOW, M.A., B.Sc.,GOLD MEDALLIST IN MATHEMATICS AT LONDON M.A.,SIXTH WRANGLER, AND FIRST CLASS FIRST DIVISION PART II. MATHEMATICAL

TRIPOS, CAMBRIDGE,AND

GK H. BBYAN, D.So., M.A., F.E.S.,SMITH'S PRIZEMAN, LATE FELLOW OK ST. PETER'S COLLEGE, CAMBRIDGE,

JOINT AUTHOR OF " COORDINATE GEOMETRY, PART I.," " THE TUTORIAL ALGEBRA,ADVANCED COURSE," ETC.

Third Impression (Second Edition).

LONDON: W. B. OLIVE,(University Correspondence College Press],

13 BOOKSELLEB.S Row, STKAND, W.C.1900.


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pPREFACE TO THE FIRST EDITION.

FOR some time past it has been felt that a gap existed betweenthe many excellent popular and non-mathematical works on As-tronomy, and the standard treatises on the subject, which involvehigh mathematics. The present volume has been compiled withthe view of filling this gap, and of providing a suitable text-bookfor such examinations as those for the B.A. and the B.Sc. degrees ofthe University of London.It has not been assumed that the reader's knowledge of mathe-matics extends beyond the more rudimentary portions of Geometry,Algebra, and Trigonometry. A knowledge of elementary Dynamicswill, however, be required in reading the last three chapters, butall dynamical investigations have been left till the end of the book,thus separating dynamical from descriptive Astronomy.The principal properties of the Sphere required in Astronomyhave been collected in the Introductory Chapter ; and, as it isimpossible to understand Kepler's Laws without a slight knowledgeof the properties of the Ellipse, the more important of these havebeen collected in the Appendix for the benefit of students who havenot read Conic Sections.

All the more important theorems have been carefully illustratedby worked-out numerical examples, with the view of showing howthe various principles can be put to practical application. Theauthors are of opinion that a far sounder knowledge of Astronomycan be acquired with the help of such examples than by learningthe mere bookwork alone.Feb. 1st, 1892.

PREFACE TO THE SECOND EDITION.

THE first edition of Mathematical Astronomy having run out ofprint in less than eight months, we have hardly considered itadvisable to make many radical changes in the present edition.We have, however, taken the opportunity of adding several notes atthe end, besides answers to the examples, which latter will, wehope, prove of assistance, especially to private students ; our readerswill also notice that the book has been brought up to date by theinclusion of the most recent discoveries. At the same time wehope we have corrected all the misprints that are inseparable froma first edition. Our best thanks are due to many of our readers fortheir kind assistance in sending us corrections and suggestions.

Nov. 1st, 1892.


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CONTENTS.INTRODUCTORY CHAPTER. PAOB

ON SPHERICAL GEOMETRY iDefinitions iiProperties of Great and Small Circles iiiOn Spherical Triangles v

CHAPTER I.THE CELESTIAL SPHERE./Sect. I. Definitions Systems of Coordinates 1

II. The Diurnal Rotation of the Stars 13III. The Sun's Annual Motion in the EclipticThe Moon's Motion Practical Applications 20

CHAPTER II.THE OBSERVATORY.Sect. I. Instruments adapted for Meridian Observations 35

II. Instruments adapted for Observations off theMeridian 54CHAPTER III.THE EARTH.

Sect. I. Phenomena depending on Change of Positionon the Earth 63II. Dip of the Horizon 73

III. Geodetic Measurements Figure of the Earth 77CHAPTER IV.

THE SUN'S APPARENT MOTION IN THE ECLIPTIC.Sect. I. The Seasons 87

II. The Ecliptic 99III. The Earth's Orbit round the Sun 105

CHAPTER V.ON TIME.^/Sect. I. The Mean Sun and Equation of Time 115

II. The Sun-dial 125III. Units of Time The Calendar 127IV. Comparison of Mean and Sidereal Times 129


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CONTENTS.CHAPTER VI.

PACKATMOSPHERICAL REFRACTION AND TWILIGHT 140CHAPTER VII.

THE DETERMINATION OF POSITION ON THE EARTH.Sect. I. Instruments used in Navigation 153^X, II. Finding the Latitude by Observation 102^ HI. To find the Local Time by Observation 171IV. Determination of the Meridian Line 175CXJ, V. Longitude by Observation 177VI. Captain Sumner's Method 187

CHAPTER VIII.THK MOON.Sect. I. Parallax The Moon's Distance and Dimensions 191

II. Synodic and Sidereal Months Moon's PhasesMountains on the Moon 200III. The Moon's Orbit and Rotation 209

CHAPTER IX.ECLIPSES.

Sect. I. General Description of Eclipses 219,, II. Determination of the Frequency of Eclipses 224III. Occultations Places at which a Solar Eclipse

is visible 232

CHAPTER X.THE PLANETS.Sect. I. General Outline of the Solar System ... ... 238

II. Synodic and Sidereal Periods Description ofthe Motion in Elongation of Planets, asseen from the Earth Phases 244III. Kepler's Laws of Planetary Motion 253IV. Motion relative to Stars Stationary Points ... 258V. Axial Rotations of Sun and Planets 264

CHAPTER XLTHE DISTANCES OF THE SUN AND STARS.

Sect. I. Introduction Determination of the Sun'sParallax by Observations of a SuperiorPlanet at Opposition 267

II. Transits of Inferior Planets 271,, III. Annual Parallax, and Distances of the Fixed

Stars 283IV. The Aberration of Light ... 293


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CONTENTS.

DYNAMICAL ASTRONOMY.CHAPTER XII. PAORTHE ROTATION OF THE EARTH 315CHAPTER XIII.

THE LAW OP UNIVERSAL GRAVITATION.Sect. I. The Earth's Orbital Motion Kepler's Lawsand their Consequences 337II. Newton's Law of Gravitation Comparison ofthe Masses of the Sun and Planets 352

III. The Earth's Mass and Density 362CHAPTER XIV.

FURTHER APPLICATIONS OF THE LAW OF GRAVITATION.Sect. I. The Moon's Mass Concavity of Lunar Orbit... 371

II. The Tides 375,, III. Precession and Nutation 392IV. Lunar and Planetary Perturbations 406

NOTES.Diagram for Southern Hemisphere 421The Photochronograph 421Effects of Dip, &c., on Rising and Setting 422

APPENDIX.Properties of the Ellipse 423Table of Constants 426

ANSWERS TO EXAMPLES AND EXAMINATION QUESTIONS 428INDEX 434


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INTRODUCTORY CHAPTER,ON SPHERICAL GEOMETRY.

Properties of the Sphere which will be referred to in the course of theText.(1) A Sphere may be defined as a surface all points on which areat the same distance from a certain fixed point. This point is theCentre, and the constant distance is the Radius.(2) The surface formed by the revolution of a semicircle aboutits diameter is a sphere. For the centre of the semicircle is keptfixed, and its distance from any point on the surface generated willbe equal to the radius of the semicircle.

FIG. 1.(3) Let PqQP' be any position of the revolving semicircle whosediameter PP' is fixed. Let OQ be the radius perpendicular to PP',Cq any other line perpendicular to PP', meeting the semicircle in

q. (We may suppose these lines to be marked on a semicircular disc ofcardboard.) As the semicircle revolves, the lines OQ, Cgwill sweepout planes perpendicular to PP', and the points Q, q will trace outin these planes circles HQRK, hqrJc, of radii OQ, Cq respectively.From this it may readily be seen that Every plane section of asphere is a circle,

4-STKON, 5


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ii ASTRONOMY.Definitions.

(4) A great Circle of a sphere is the circle in which it is cut byany plane passing through the centre (e.g., HQRK, PqQP' or PrRP ).A small circle is the circle in which the sphere is cut by any planenot passing through the centre (e.g., hqrk).(5) The axis of a great or small circle is the diameter of the

sphere perpendicular to the plane of the circle. The poles of thecircle are the extremities of this diameter. (Thus, the line PP isthe axis, and P, P' are the poles of the circles HQK and hqJc).(6) Secondaries to a circle of the sphere are great circles passingthrough its poles. (Thus, PQP' and PRP" are secondaries of the

circles HQK, hqk).

FIG. 2.(7) The angular distance between two points on a sphere ismeasured by the arc of the great circle joining them, or by the anglewhich this arc subtends at the centre of the sphere. Thus, the dis-tance between Qand Bis measuredeither by thearc QE, or by the angleQOR. Since the circular measure of L QOR = (arc Qft) -f- (radius of

sphere), it is usual to measure arcs of great circles by the angleswhich they subtend at the centre. This remark does not apply tosmall circles.(8) The angle between two great circles is the angle betweentheir planes. Thus, the angle between the circles PQ, PR is the anglebetween the planes PQP', 7'EP'. It is called "the angle QPR."(9) A spherical triangle is a portion of the spherical surfacebounded by three arcs of gr.eat_circles. Thus, in Fig. 2, PQR is a

spherical triangle, but Pqr is not a spherical triangle, because qr isnot an arc of a great circle. We may, however, draw a great circlepassing through q and r, and thus form a spherical triangle Pqr.


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SPHERICAL GEOMETRY. illProperties of Great and Small Circles.

(10) All points on a small circle are at a constant (angular))distance from the pole.For, as the generating semicircle revolves about PP7 , carrying galong the small circle hk, to r, the arc Pq = arc Pr, and Z POq = L POr.The constant angular distance Pq is called the spherical, orangular radius of the small circle. The pole P is analogous to thecentre of a circle in plane geometry.(11) The spherical radius of a great circle is a quadrant, or,All points on a great circle are distant 90 from its poles.For, as Q, by revolving about PP', traces out the great circle

HQRK, we have L POQ = L POR = 90, and therefore, PQ, PE arequadrants.(12) Secondaries to any circle lie in planes perpendicular tothe plane of the circle.For PP' is perpendicular to the planes of the circles HQK, liqk,therefore any plane through PP/, such as PQP' or PEP', is also per-

pendicular to them.(13) Circles which have the same axis and poles lie in parallel,

planes. For the planes HQK, hqk are parallel, both being perpen-dicular to the axis PP'. Such circles are often called parallels.(14) If any number of circles have a common diameter, their

poles all lie on the great circle to which they are secondaries, andthis great circle is a common secondary to the original circles.For if OA is the axis of the circle PQP', then OA is perpendicular-to POP'. Hence, if the circle PQP7 revolves about PP', A traces out.the great circle HQRK, of which P, P7 are poles. We likewise see that

(15) If one great circle is a secondary to another, the latter isalso a secondary to the former.This is otherwise evident, since their planes are perpendicular.(16) The angle between two great circles is equal to

(i.) The angle between the tangents to them at their pointsof intersection ;(ii.) The arc which they intercept on a great circle to whichthey are both secondaries ;

(iii.) The angular distance between their poles.Let Ft, Pu be the tangents at P to the circles PQ, PE, and let A, Bbo the poles of the circles. If we suppose the semicircle PQP' torevolve about PP' into the position PEP', the tangent at P willrevolve from Pt to Pu, the radius perpendicular to OP will revolvefrom OQ to 07?, and the axis will revolve from OA to OB. All theselines will revolve through an angle equal to the angle betweenthe planes PQP', PRP/ , and this is the angle QPE between thecircles (Def. 8). BLenee,

le between circles PQ, PR = L tPu = L QOR


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{y ASTEONO^TT.

(17) The arc of a small circle subtending a given angle at thepole is proportional to the sine of the angular radius.Let qr be the arc of the small circle hqrJc, subtending L qPr at P,and let G be the centre of the circle. Evidently L qCr

= L QOR(since Cq, Gr are parallel to OQ, OB). Hence, the arcs qr, QR areproportional to the radii Cq, OQ,

. arc qr = G = Gq_ = ghl pQq = gin p^arc QR OQ OqBut QR is the arc of a great circle subtending the same angle at thepole P hence the arc qr is proportional to sin Pq, as was to be shown.Since qQ = 90 - PQ, therefore sin Pq - cos gQ, so that the arc qr isproportional to the cosine of the angular distance of the small circle(jr from the parallel great circle QR.

FIG. 3. FIG. 4.

(18) Comparison of Plane and Spherical Geometry.It may be laid down as a general rule that great circles and smallcircles on a sphere are analogous in their respective properties to

straight lines and circles in a plane. Thus, to join two points on asphere means to draw the great circle passing through them.

Secondaries to a great circle of the sphere are analogous to per-pendiculars on a straight line. The distance of a point from anygreat circle is the length of the arc of a secondary drawn from thepoint to the circle. Thus, rR is the distance of the point r from thegreat circle HQRK.


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SPHEEICAL GEOMETftf. VOn Spherical Triangles*

(19) Parts of a Spherical Triangle. A spherical triangle, like aplane triangle, has six parts, viz., its three sides and its three angles.The sides are generally measured by the angles they subtend at thecentre of the sphere, so that the six parts are all expressed as angles.Any three given parts suffice to determine a spherical triangle,but there are certain " ambiguous cases " when the problem admitsof more than one solution. The formulge required in solvingspherical triangles form the subject of Spherical Trigonometry,and are in every case different from the analogous formulaj in PlaneTrigonometry. There is this further difference, that a sphericaltriangle is completely determined if its three angles are given.

Thus, two spherical triangles will, in general, be equal if theyhave the following parts equal :(i.) Three sides.

(ii.) Two sides andincluded angle.(iii.) Two sides and one opposite

angle.

(iv.) Three angles,(v.) Twoanglesandadjacentside.(vi.) Two angles and one opposite

side.Cases (iii.) and (vi.) may be ambiguous.

(20) Right-angled Triangles. If one of the angles is a rightangle, two of the remaining five parts will determine the triangle.

(21) Triangle with two right angles. The properties of aspherical triangle, such as PQR, Fig. 3, in which one vertex P isthe pole of the opposite side QR, are worthy of notice. Here twoof the sides, PQ, PR, are quadrants, and two angles Q, R are rightangles. The third side QR is equal to the opposite angle QPR,

(22) Triangle with, three right angles. If, in addition, the angleQPR is a right angle (Fig. 4), QR will be a quadrant. The trianglePQR will, therefore, have all its angles right angles, and all its sidesquadrants, and each vertex will be the pole of the opposite side.The planes of the great circles forming the sides, are three planesthrough the centre mutually at right angles, and they divide thesurface of the sphere into eight of these triangles ; thus the area ofeach triangle is one-eighth of the surface of the sphere.

(23) The three angles of a spherical triangle are togethergreater than two right angles.[For proof, see any text-book on Spherical Geometry.]

(24) If the sides of a spherical triangle, when expressed as angles,are very small, so that its linear dimensions are very small com-pared with the radius of the sphere, the triangle is very approxi-mately a plane triangle.

Thus, although the Earth's surface is spherical, a triangle whosesides are a few yards in length, if traced on the Earth, will not bedistinguishable from a plane triangle. If the sides are severalmiles in length, the triangle will still be very nearly plane.


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vi AJSTKONOMY.

(25) Any two sides 6f a spherical triangle are togethergreater than the third side. For if we consider the plane angleswhich the sides subtend at the centre of the sphere, any two ofthese are together greater than the third, by Euclid XL, 20.

(26) The following application of (25) is of great use in astronomy,and is analogous to Euclid III., 7, 8.Let AHBK be any given great or small circle whose pole is P,Zany other given point on the sphere, and let the great circle ZPmeet the given circle in the points A, B. Then A, B are the twopoints on the given circle whose distances from Z are greatest andleast respectively.For let H be any other point on the circle. Join ZH, HP.Then, in spherical A ZPH, ZP + PH> ZH. But PH = PA ;

/. ZP + PA > ZH,i.e., ZA>ZH.

Also, if Z is on the opposite side of the circle to P, thenZII+PH>PZ', .:ZH + PB>PZ; .:ZH>PZ-PB,i.e., ZH>ZB.

If Z' be a point on the same side of the circle as P, then PZ' + Z'H>PH. But PH - PB. .'. PZ'-t Z'H^PB..-. Z'H>PB-PZ',

i.e., Z'H>Z'B, as before.lienee, A is further from Z, Z', and B is nearer to Z, Z', than anyother point on the circle.(27) If H, K are the two points on the circle equidistant from Z,the spherical triangles ZPH, ZPK have ZP common, ZH = ZK (by

hypothesis^), and PH = PK [by (10)], hence they are equal in allrespects ; thus L ZPH = L ZPK, and L PZH = L PZK.Hence PH, PK are equally inclined to PB, as are also ZH, ZK.Similar properties hold in the case of the point Z'. These pro-perties are of frequent uw.


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ASTRONOMY.CHAPTEE I.

THE CELESTIAL SPHERE.SECTION I. Definitions Systems of Co-ordinate*.

1 . Astronomy is the science which deals with the celestialbodies. These comprise all the various bodies distributedthroughout the universe, such as the Earth (considered as awhole),

the Sun, the Planets, the Moon, the comets, the fixedstars, and the nebulae. It is convenient to divide Astronomyinto three different branches.The first may be called Descriptive Astronomy. It isconcerned with observing and recording the motions of the

various celestial bodies, and with applying the results ofsuch observations to predict their positions at any subsequenttime. It includes the determination of the distances, and themeasurement of the dimensions of the celestial bodies.The second, or Gravitational Astronomy, is an appli-cation of the principles of dynamics to account for the motionsof the celestial bodies. It includes the determination of theirmasses.The third, called Physical Astronomy, is concernedwith determining the nature, physical condition, temperature,and chemical constitution of the celestial bodies.The first branch has occupied the attention of astronomersin all ages. The second owes its origin to the discoveries of

Sir Isaac Newton in the seventeenth century ; while thethird branch has been almost entirely built up in the presentcentury.In this book we shall treat exclusively of Descriptive andGravitational Astronomy.


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ASTRONOMY.: -;2: :The ;C.elesti.al Sphere. On observing the stars it is' not^ 'difficult to imagine that they are bright points dottedabout on the inside of a hollow spherical dome, whose centreis at the eye of the observer. It is impossible to form anydirect conception of the distances of such remote bodies ; allwe can see is their relative directions. Moreover, mof-tastronomical instruments are constructed to determine onlythe directions of the celestial bodies. Hence it is importantto have a convenient mode of representing directions.

FIG. 6.The way in which this is done is shown in Figure 6. Letbe the position of any observer, A, , C, &c., any stars orother celestial bodies. About 0, as centre, describe a spherewith any convenient length as radius, and let the lines joiningto the stars A, J3, C meet this sphere in a, ft, c respectively.Then the points a, I, c will represent, on the sphere, the

directions of the stars A, H, C, for the lines joining thesepoints to will pass through the stars themselves. In thismanner we obtain, on the sphere, an exact representation ofthe appearance of the heavens as seen from 0. Such asphere is called the Celestial Sphere.This sphere may be taken as the dome upon which the starsappear to lie. But it must be carefully borne in mind thatthe stars do not actually lie on a sphere at all, and that theyare only so represented for the sake-of convenience.


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THE CELESTIAL SPHERE.3. Use of the Globes. The representation of directions

of stars by points on a sphere is well exemplified in the old-fashioned star globes. Such a globe may be used as theobserver's celestial sphere ; but it must be remembered thatthe directions of the stars are the lines joining the centre tothe corresponding points on the sphere ; for in every case theobserver is supposed to be at the centre of the celestialsphere.The properties given in the Introduction on Spherical Geo-metry are applicable to the geometry of the celestial sphere.A knowledge of thorn will be assumed in what follows.

4. Angular Distances and Angular Magnitudes.Any plane through the observer will be represented on thecelestial sphere by a great circle. The arc of the great circleab (Fig. 6) represents the angle a 01 or A OB which the starsA, subtend at 0. This angle is generally measured indegrees, minutes, and seconds, and is called the angulardistance between the stars. This angular distance mustnot be confused with their actual distance AB. In the sameway, when we are dealing with a body pf perceptible dimen-sions, such as the Sun or Moon (DF, Fig. 6), we shall defineits angular diametsr as the angle DOF, subtended by adiameter at the observer's eye. This angular diameter ismeasured by the arc df of the celestial sphere, that is, by thediameter of the projection of the body on the celestial sphere.From the figure it is evident that

Od 01)'Since DF is the actual linear diameter of the body, mea-sured in units of length, the last relation shows us that the

angular diameter (df) of a body varies directly as its lineardiameter DF, and inversely as OD, the distance of the bodyfrom the observer's eye.As the eye can only judge of the dimensions of a bodyfrom its angular magnitude, this result is illustrated by the1'act that the nearer an object is to the eye the larger it looks,and vice versd. Thus, if the distance of the object be doubled,it will only look half as large, as may be easily verified.


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4 ASTRONOMY.5. The Directions of the Stars are very approxi-mately independent of the Observer's Position onthe Earth.This is simply a consequence of the enormously great dis-tances of all the stars from the Earth. Thus,let x (Fig. 7) denote any star or other celestial

body, S, JZtwo different positions o^ the observer.If the distance SJ be only a very small fractionof the distance Sx, the angle Ex8 will be verysmall, and this angle measures the difference be-tweenthe directions of x as seenfrom ^and from 8.In illustration, if we observe a group of objectsa mile or two off, and then walk a few feet in anydirection, we shall observe no perceptible change FIG. 7.in the apparent directions or relative positions of the objects.If Ex be drawn parallel to Sx, the angle xEx will beequal to ExS, and will therefore be very small indeed.Hence, Ex will very nearly coincide in direction with Ex'.Thus, considering the vast distances of the stars, we see thatThe lines joining a Star to different points of theEarth may be considered as parallel.*The stars will, therefore, always be represented by thesame points on a star globe, or celestial sphere, no matterwhat be the position of the observer. The great use of thecelestial sphere in astronomy depends on this fact.

6. Motion of Meteors. The projection of bodies on thecelestial sphere is well illustrated by the apparent motionof a swarm of meteors. Where such a swarm is movinguniformly, all the meteors describe (approximately) parallelstraight lines. II we draw planes through these lines andthe observer, they will intersect in a common line, namely,the line through the observer parallel to the direction of thecommon motion of the meteors. The planes will, therefore,cut the celestial sphere in great circles, having this line astheir common diameter. These great circles represent theapparent paths >i (he meteors on the celestial sphere. Thepaths appear, therefore, to radiate from a common point,namely, one of the extremities of this diameter.This point is called the Radiant, and by observing itsposition the direction of motion of the meteors is determined.

* This is not true in the case of the Moon.


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tHE CELESTIAL StHE&E. 67. Zenith and Nadir. Horizon. If, through the

observer, a line be drawn in the direction in which gravityacts(i.e., the direction indicated by a plumb-line), it will meetthe celestial sphere in two points. One of these is verticallyabove the observer, and is called the Zenith; the other is

vertically below the observer, and is called the Nadir. (Fig.6, and Z, N, Fig. 8.)If the plane through the observer parallel to the surfaceof a liquid at rest be produced, it will cut the celestialsphere in a great circle. This great circle is called theCelestial Horizon. (Fig. 6, and sEnW, Fig. 8.)It is proved in Hydrostatics that the surface of a liquid atrest is a plane perpendicular to the direction of gravity.Hence, the celestial horizon is the great circle whose polesare the zenith and nadir. "We might have defined thehorizon by this property.From the above definition, it is evident that, to an observerwhose eye is close to the surface of the ocean, the celestialhorizon forms the boundary of the visible portion of thecelestial sphere. On land, however, the boundary, or visiblehorizon (as it is called), is always more or less irregular,owing to trees, mountains, and other objects.

8. Diurnal Motion of the Stars. If we observe thesky at different intervals duringthe night, we shall find that thestars always maintain the sameconfigurations relative to oneanother, but that their actualsituations in the sky, relative tothe horizon, are continuallychanging. Some stars will setin the west, others will rise inthe east. One star which issituated in the constellation calledthe l< Little Bear," remains almost FlG - 8 -fixed. This star is called Polaris, or the Pole Star. All theother stars describe on the celestial sphere small circles(Fig. 8) having a common pole P very near the Pole Star,and the revolutions are performed in the same period of time,namely, about 23 hours 56 minutes of our ordinary time.


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6 ASTEONOMt.9. Celestial Poles, Equator, and Meridian. Thecommon motion of the stars may most easily be conceived by

imagining them to be attached to the surface of a spherewhich is made to revolve uniformly about the diameter PP'.The extremities of this diameter are called the CelestialPoles. That pole, P, which is above the horizon in northernlatitudes is called the North Pole, the other, P\ is calledthe South Pole.The great circle, JEQR W, having these two points for itspoles, is called the Celestial Equator. It is, therefore, thecircle which would be traced out by the diurnal path of astar distant 90 from either pole.

The Meridian is the great circle (PZP'N, Fig. 9) passingthrough the zenith and nadir and the celestial poles. It cutsboth the horizon and equator at right angles [by Spher.Geom. (12), since it passes through their poles].


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THE CELESTIAL SPHEKE. 710. The Cardinal Points. The East and West

Points (J, W, Eig. 9) are the points of intersection of theequator and horizon. The North and South Points(&, S) are the intersections of the meridian with the horizon.Verticals. rSecondaries to the horizon, i.e., great circlesthrough the zenith and nadir., are called Vertical Circles,or, briefly, Verticals. Thus, the meridian is a vertical.The Prime Vertical is the vertical circle (ZENTF) passingthrough the east and west points.Since P is the pole of the circle QERW, and ^is the poleof nEsWy therefore E, W are the poles of the meridianPZP'N. Hence the horizon, equator, and prime verticalwhich pass through E, W, are all secondaries to the meridian ;they therefore all cut the meridian at right angles.

11. Annual Motion of the Sun. The Ecliptic.The Sun, while participating in the general diurnal rotationof the heavens, possesses, in addition, an independentmotion of its own relative to the stars.Imagine a star globe worked by clockwork so as to revolveabout an axis pointing to the celestial pole in the same peri-odic time as the stars. On such a moving globe the directionsof the stars will always be represented by the same points.During the daytime let the direction of the Sun be marked onthe globe, and let this process be repeated every day for a year.We shall thus obtain on the globe a representation of theSun's path relative to the stars, and it will be found that

(i.) The Sun moves from west to east, and returns to thesame position among the stars in the period called a year ;(ii.) The relative path on the celestial sphere is a great

circle, inclined to the equator at an angle of about 23 27f.This great circle (CTL ===, Fig. 9) is called the Ecliptic."We may, therefore, briefly define the ecliptic as the greatcircle which is the trace, on the celestial sphere, of the Sun'sannual path relative to the stars.The intersections of the ecliptic and equator are calledEquinoctial Points. One of them is called the FirstPoint of Aries ; this is the point through which the Sunpasses when crossing from south to north of the equator, andit is usually denoted by the symbol T The other is calledthe First Point of Libra, and is denoted by the symbol =0=,


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ASTKONOMY.12. Coordinates. In Analytical Geometry, the positionof a point in a plane is denned by two coordinates. In like

manner, the position of a point on a sphere may be denned bymeans of two coordinates. Thus, the position of a place onthe Earth is denned by the two coordinates, latitude andlongitude. For fixing the positions of celestial bodies, thefollowing different systems of coordinates are used.

13. Altitude or Zenith Distance and Azimuth. LetFig. 10 represent the celestial sphere, seen from overhead, andlot x be any star. Draw the vertical circle ZxX. Then theposition of x may be defined by either of the following pairsof coordinates, which are analogous to the Cartesian andpolar coordinates of a point in a plane respectively :

(a) The arc sX and the arc Xx ;(b) The arc Zx and the angle sZx.

Practically, however, the two systems are equivalent ; for,since Z is the pole of sX, ZX = 90, thereforeZx = 90 xXj and angle sZx = arc sX,

FIG. 10.The Altitude of a star (Xx} is its angular distance fromthe horizon, measured along a vertical.The Zenith Distance (abbreviation, Z.D.) is its angular

distance from the zenith (Zx) , orthe complement of the altitude.The Azimuth (sX or sZx) is the arc of the horizon inter-cepted between the south point and the vertical of the star,or the angle which the star's vertical makes with the meridian


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THE CELESTIAL SPHERE. 9*14. Points Of the Compass. In practical applications of Astro-nomy to navigation, it is usual to measure the azimuth in "points"and " quarter points " of the compass. The dial plate of a mariner's

compass is divided into 32 points, by repeatedly bisecting the rightangles formed by the directions of the four cardinal points. Thuseach point represents an angle of Hi degrees. The points are againsubdivided into " quarter points " of 2\ degrees. Starting from thenorth and going round towards the east, the various points are denotedas follows :N., N. byB., N.N.E., N.E. by N., N.E., N.E. by E., E.N.E., E. by N.E., E. byS., E.S.E., S.E. by E., S.E., S.E. by S., S.S.E., S. by E.S., S. by W. S.S.W., S.W. by S., S.W., S.W: by W., W.S.W , W. by S.W., W. by N., W.N.W. N.W. by W., N.W., N.W. by N., N.N.W., N. by W.The quarter points are denoted thus : E.N.B. E. means one

quarter point to the eastward of E.N.E., that is, 6 points, or70 18' 45", from the north point, taken in an easterly direction.So, too, S.S.W. W. meafli 2J points, or 28 7' 30' , measured fromthe south point westwards.15. Polar Distance, or Declination, and Hour Angle.From the pole P, draw through x the great circle PxM-, this

circle is a secondary to the equator EQ, W.Then we may take for the coordinates of x the arc Px andthe angle sPx. Or we may take the arc x3f, which is thecomplement of Px, and the arc QM, which = angle QPx.The North Polar Distance of a star (abbreviation,N.P.D.) is its angular distance (Pa;) from the celestial pole.The Declination (abbreviation, Decl.) is the angulardistance from the equator (xM), measured along a secondary,and is, therefore, the complement of the N.P.D.The great circle PxM through the pole and the star iscalled the star's Declination Circle.The Hour Angle of the star (ZPx] is the angle whichthe star's declination circle makes with the meridian.The declination may be considered positive or negative,

according as the star is to the north or south of the equator,but it is more usual to specify this by the letter N. or S., asthe case may be, and this is called the name of the declination.The hour angle is generally measured from the meridiantowards the west, and is reckoned from to 360.Either the declination and hour angle or the N.P.D. andhour angle may be taken as the two coordinates of a star.


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10 ASTBONOHY.16. Declination and Right Ascension. The position

of a celestial body is, however, more frequently defined byits declination and right ascension.'The declination has been already defined, in 15, as theangular distance of the star from the equator, measured alonga secondary. (xM, Fig. 11.)The Right Ascension (E.A.) is the arc of the equatorintercepted between the foot of this secondary and the FirstPoint of Aries. Thus, T^, Fig. 11, is the E.A. of the star a:.The E.A. of a star is always measured from T eastwardsreckoning from to 360. Thus the star w Piscium, whosedeclination circle cuts the equator 1 34' 18" west of T, hasthe E.A. 360 1 34' 18", or 358 25' 42".

FIG. 11.17. Celestial Latitude and Longitude. The position

of a celestial body may also be referred to the ecliptic insteadof the equator.The Celestial Latitude is the angular distance of thetody from the ecliptic, measured along a secondary to theecliptic. (Hx, Pig. 11.)The Celestial Longitude is the arc of the ecliptic inter-cepted between this secondary and the first point of Aries,measured eastwards from T- (T#, Pig. 11.)


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tflE CELESTIAL SPHERE. ll18. Latitude of the Observer. The celestial latitude

and longitude, defined in the last paragraph, must not beconfounded with the latitude and longitude of a place on theEarth, as there is no connection whatever between them.The Latitude of a place is the angular distance of itszenith from the equator, measured along the meridian.Thus, in Pig. 1 1 , ZQ, is the latitude of the observer.Since PQ nZ 90 ; .-. ZQ = nP, or in other words,The latitude of a place is the altitude of the Celestial Pole.The complement of the latitude is called the Colatitude.Hence, in Pig. 11, PZ is the colatitude of the observer,and is the angular distance of the zenith from the pole.In this book the latitude of an observer will generally bedenoted by the symbol /, and the colatitude by c.The longitude of a place will be defined in Chapter III.19. Obliquity of the Ecliptic. The inclination of the

ecliptic to the equator is called the Obliquity. In Pig. 11,Q T C is the obliquity. As stated in 1 1 , this angle is about23 27-'. We shall generally denote the obliquity by i.20. Advantages of the Different CoordinateSystems. The altitude and azimuth of a celestial bodyindicate its position relative to objects on the Earth. Owing,however, to the diurnal motion, they are constantly changing.The N.P.D. and hour angle also serve to determine thestar's

position relative to the earth, and have this furtheradvantage, that the N.P.D. is constant, while the hour angleincreases at a uniform rate.

Since the equator and first point of Aries partake of thecommon diurnal motion of the stars, the declination and rightascension of a star are constant. These coordinates are, there-fore, the most suitable for tabulating the relative positions ofthe various stars on the celestial sphere.The celestial latitude and longitude of a celestial body arealso unaffected by the diurnal motion. They are most useful in

defining the positions of the Sun, Moon, planets, and comets,for the first always moves in the ecliptic, while the pathsdescribed by the others are always very near the ecliptic.

21. Recapitulation. Por the sake of convenient refer-ence, we give on the next page a list of all the definitions ofthis chapter, with references to Pigs. 11, 12.ASTRON. c


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12 ASTRONOMY.GREAT CIRCLES.

Horizon, nEsW.Equator, EQWR.Meridian, ZsZ'n.Prime Vertical, ZEZ'W.

THEIR POLES.Zenith, Z-, Nadir, Z '.North Pole, P ; South Pole, P.East Point, E\ West Point, W.NorthPoint, n ; South Point, s.

Ecliptic, T i:Z ; Equinoctial Points, T, =2=, viz. : EirstPoint of Aries, T , and Eirst Point of Libra, b ; Yertical ofStar, ZxX-, Declination Circle of Star, Pxlf.

FIG. 12.COORDINATES.

Altitude, Xx ; '")or Zenith Distance, Zx. )North Polar Distance, Px.

Declination, MX.Celestial Latitude, Hx.

Azimuth, sX = sZx.Hour Angle, QM= ZPx.Bight Ascension, T^Celestial Longitude,OTHER ANGLES. Obliquity of Ecliptic (t) CT Q-Observer's Latitude (1) = ZQ = nP. Colatitude (c) = PZ.Notice that the circles on the remote side of the celestial sphereare dotted.


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CELESTIAL SPHEKE. 13SECTION II. The Diurnal Rotation of the Stars.

22. Sidereal Day and Sidereal Time. A SiderealDay is the period of a complete revolution of tlie stars aboutthe pole relative to the meridian and horizon. Like thecommon day it is divided into 24 hours (h.), and these aresubdivided into 60 minutes (m.) of 60 seconds (s.) each.The sidereal day commences at "Sidereal Noon," i.e., theinstant when the first point of Aries crosses the meridian.The Astronomical Clock, which is the clock used in

observatories, indicates sidereal time. The hands shouldindicate Oh. Om. Os. when the first point of Aries crosses themeridian, and the hours are reckoned from Oh. up to 24h.,when T again comes to the meridian and a new day begins.From the facts stated in 8, it appears that the siderealday is about 4 minutes shorter than the ordinary day. Thestars are observed to revolve about the pole at a perfectlyuniform rate, so that the sidereal day is of invariable length,and the angles described by any star about the pole are pro-portional to the times of describing them. Thus, the hourangle of a star (measured towards the west) is proportionalto the interval of sidereal time that has elapsed since the starwas on the meridian.Now, in 24 sidereal hours the star comes round again tothe meridian, after a complete revolution, the hour anglehaving increased from to 360. Hence the hour angle in-

creases at the rate of 15 per hour. Hence, also, it increases15' per minute, or 15" per second.The hour angle of a star is, for this reason, generallymeasured by the number of hours, minutes, and seconds ofsidereal time taken to describe it. It is then said to beexpressed in time. Thus,The hour angle of a star, when expressed in time*

is the interval of sidereal time that has elapsedsince the star was on the meridian.In particular, since the instant when T is on the meridian

is the commencement of the sidereal day, we see thatThe sidereal time is the hour angle of the firstpoint of Aries when expressed in time.


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14 ASTHONOMY.23. To reduce to angular measure any angle ex-pressed in time. Multiply ~by 15. The hours, minutes, and

seconds of time will thus be reduced to degrees, minutes, andseconds of angle.Conversely, to reduce to time from angular measurewe must divide by 15, and for degrees, minutes, and seconds,write hours, minutes, and seconds.EXAMPLES. 1. To find, in angular measure, the hour angle of astar at 15h. 21m. 50s. of sidereal time after its transit. The processstands thus

15 21 50

230 27 30/. the angular measure of the hour angle is 230 27' ?0"2. To find the sidereal time required to describe 230 27' 30"(converse of Ex. 1).15 ) 230 27 30

15 21 50 ; .-. required time= 15h. 21m. 50s.

24. Transits. The passage of the star across the meri-dian is called its Transit.

Let x be the position of any star in transit (Fig. 13).The star's E.A. = T Q or rPQ = hour angle of T= sidereal time expressed in angle.Hence, the right ascension of a star, when ex-

pressed in time, is equal to the sidereal time of itstransit.In practice the R.A. of a star is always expressed in time.

Thus, the R.A. of a Lyrse is given in the tables aa18h. 33m. 14-8s., and not as 278 18' 42".


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THE CELESTIAL SPHEEE. 15Again, let 2 be the meridian zenith distance Zx, considered

positive if the -star transits north of the" zenith, d the star'snorth declination Qx, and I the north latitude QZ. Wohave evidently - Qx = QZ+Zx;d = i+*cor (star's N. decl.)= (lat. of observer)+ (star's meridian Z.D.)

This formula will hold universally if declination, latitude,and zenith distance are considered negative when south.Hence the R.A. and decl. of a star maybe found by observingits sidereal time of transit and its meridian Z.D., the latitude ofthe observatory being known.

Conversely, if the R.A. and decl. of a star are known, wecan, by observing its time of transit and meridian Z.D., deter-mine the sidereal time and the latitude of the observatory.By finding the sidereal time we may set the astronomicalclock.A star whose E.A. and decl. have been tabulated, is called

a known star.In Chapter II. we shall describe an instrument called the

Transit Circle, which is adapted for observing the times oftransit and meridian zenith distances of celestial bodies.

25. General Relation between R.A. and hourangle. Let xl (Fig. 13) be any star not on the meridian.Then

z QpXl = L QPr- t rP^ = ^ QPr rM]hence, if angles are expressed in time,(star's hour angle) = (sidereal time) (star's H.A.).

Hence, given the 11.A. and decl. of a star, we can find its hourangle and N.P.D. at any given sidereal time, and by this meansdetermine the star's position on the'observer's celestial sphere.Or we can construct the star's position thus On the equator,in the westward direction from Q, measure off QT equal tothe sidereal time (reckoning 15 to the hour). Prom T east-wards, measure fM equal to the star's It.A.; and from 3f, inthe direction of the pole, measure off Mxl equal to the star'sdeclinatiqn. We thus find the star xr


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1 6 ASTRONOMY.*26. Transformations. If the R.A. and decl. of a star are

given, its celestial latitude and longitude may be found, and viceversti ; but the calculations require spherical trigonometry. Theprocess is analogous to changing the direction of the axes throughan angle i, in plane coordinate geometry. Again, the Z.D. andazimuth may be calculated from the N.F.D. and hour angle, bysolving the triangle ZPx^ We know the colatitude PZ, Px^ andL ZPx t , and we have to determine Zxi and L QZx } (= ISO PZxJ.In the last article we showed how to find the hour angle interms of the R.A., or vice versA, the sidereal time being known.Hence we see that, given the coordinates of a star referred to onesystem, its coordinates referred to any other of the systems can bocalculated at any given instant of sidereal time.

27. Culmination and Southing of Stars. A celestialbody is said to culminate when its altitude is greatest orleast.

Since the fixed stars describe circles about the pole, itreadily follows, from Spherical Geometry (26), that a starattains its greatest or least zenith distancewhenon the meridian,and, therefore, that its culmination is the same as its transit.This is not strictly the case with the Sun, because, owing toits independent motion, its polar distance is not constant ;hence it does not describe strictly a small circle about the pole.When a star transits S. of the zenith it is said to south.

28. Circumpolar Stars. A Circumpolar Star at anyplace is a star whose polar distance is less than the latitudeof the place. Its declination must, therefore, be greaterthan the colatitude.On the meridian let Px and Px' be measured, each equal tothe KP.D. of such a star (Fig. 14). Then x and x' will bethe positions of the star at its transits. Since Px < Pn, bothx' and x will be above n. Hence, during a sidereal day a cir-cumpolar star will transit twice, once above the pole (at x)and once below the pole (at x'), and both transits will bevisible. The two transits are distinguished as the upperand lower culminations respectively, and they succeed oneanother at intervals of 12 sidereal hours ( since xPx' = 180).The altitude of the star is greatest at upper, and least atlower culmination, as may easily be seen from Sph. Geom.(26) by considering the zenith distances. Hence the altitudeis never less than nx, and the star is always above the horizon.


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SinceTHE CELESTIAL SPHEBE.

nx-nP=Px = Px = nPnaf,17

that is,The observer's latitude is half the sum of the

altitudes of a circumpolar star at upper and lowerculminations.Also, Px \ (nx nx) ;that is,The Star's N.P.D. is half the difference of itstwo meridian altitudes.

These results will require modification if the upper culmi-nation takes place south of the zenith as at 8. The meridianaltitude will then be measured by sS, and not nS. Here,nS = 180 sS, and we shall, therefore, have to replace thealtitude at upper culmination by its supplement.South Circumpolar Stars. If the south polar dis-

tance of a star is less than the north latitude of the observer,the star will always remain below the horizon, and will,therefore, be invisible. Such a star is called a South Cir-cumpolar Star.

EXAMPLE. The constellation of the Southern Cross ( Crux)is invisible in Europe, for its declination is 62 30' S ; there-fore its south polar distance is 27 30', and it will, therefore,pot be visible in north latitudes higher than 27 30'.


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18 ASTBONOMY.29. Rising, Southing, and Setting of Stars. If theN. and S. polar distances of a star are both greater than the

latitude, it will transit alternately above and below thehorizon. This shows that the star will be invisible during acertain portion of its diurnal course. Astronomically, thestar is said to rise and set when it crosses the celestialhorizon.

Let J, V be the positions of any star when rising and settingrespectively.

FIG. 15.

In the spherical triangles Pnb,PI = Pb' (each being the star's KP.D.),right L Pnb = right L Pnb',and Pn is common.

Hence the triangles are equal in all respects ; thereforeZ nPb = Z nPb',

and the supplements of these angles are also equal, that is,L sPb = L sPb'.But the angle sPb, when reduced to time, measures theinterval of time taken by the star to get from b to the meri-dian, and sPV measures the time taken from the meridian tob'. Hence,The interval of time between rising and southingis equal to the interval between southing and setting.


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THE CELESTIAL SPHERE. 19Thus, if , f are the times of rising and setting, and T the

time of transit, we have T t tfT.

The time of transit is the arithmetic mean betweenthe times of rising and setting.

In order to facilitate the calculations, tables have been constructedgiving the values of T t for different latitudes and declinations.

If the observer's latitude Pn and the star's polar distance Pb areknown, it is possible (by Spherical Trigonometry) to solve the right-angled triangle PZm, and to calculate the angle nPb, and thereforealso the angle &Ps. This angle, when divided by 15, gives the timeT t. Moreover, the sidereal time of transit T is known, being equalto the star's R.A. Hence the sidereal times of rising and setting canbe found.

If the star is on the equator, it will rise at E and set at W.Since JSQWis a semicircle, exactly half the diurnal path willbe above the horizon, and the interval between rising andsetting will be 12 sidereal hours. If the star is to the northof the equator, it will rise at some point b between E and ,so that

L IPs > Z JEPs,i.e., / bPs > 90,and the star will he above the horizon for more than 12 hours.Similarly, if the star is south of the equator, it will rise at apoint c between E and *, and will be above the horizon forless than 12 hours.Prom the equality of the triangles bPn, b'Pn (Pig. 15), we

also see thatnb = nb', and sb = sb'.

Hence the diameter (ns) of the celestial sphere, joining thenorth and south points, bisects the arc (W) between thedirections of a star at rising and setting.This gives us an easy method of roughly determining, by

observation, the directions of the cardinal points ; but, owingto the usual irregularities in the visible horizon, the methoijis not very exac.


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20 ASTRONOMY.SECTION III. The Sun's Annual Motion in the EclipticThe Moon's Motion Practical Applications.

30. The Sun's Motion in Longitude, Bight Ascen-sion and Declination. In 11, we briefly describedthe Sun's apparent motion in the heavens relative to thefixed stars. "We defined a Year as the period of a completerevolution, starting from and returning to any fixed pointon the celestial sphere. The Ecliptic was defined as thegreat circle traced out by the Sun's path, and its points ofintersection with the Equator were termed the First Pointof Aries and First Point of Libra, or together, theEquinoctial Points.We shall now trace, by the aid of Pig. 16, the variationsin the Sun's coordinates during the course of a year, startingwith March 21st, when the Sun is in the first point of Aries.We shall, as usual, denote the obliquity by i, so thati = 23 27' nearly.

FIG. 16.On March 21st the Sun crosses the equator, passing

through the first point of Aries (r). This is the VernalEquinox, and it is evident from the figure thatSun's longitude = 0, B.A. = O, Decl. = 0.Prom March 21st to June 2 1st the Sun's declination isnorth, and is increasing.


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THE CELESTIAL SPHEEE. 21On June 21st the Sun has described an arc of 90 from r

on the ecliptic, and is at C (Fig. 16). This is called theSummer Solstice. If we draw the declination circlePCQ, the spherical triangleT OQ is of the kind described inSph. Geom. (21), and CP is a secondary to the ecliptic.Hence (Sph. Geom. 26) the Sun's polar distance CP is aminimum and therefore its decl. a maximum.

Also r Q = 90 and CQ = tCrQ = i. HenceSun's longitude = 90, B.A. = 90 - 6h.,

N. Decl. = /, (a maximum).From June 21 to September 23 the Sun's declination is

still north, but is decreasing.On September 23rd the Sun has described 180, and isat the first point of Libra (=), the other extremity of thecommon diameter of the ecliptic and equator. This is theAutumnal Equinox, and we haveSun's long. = 180, R.A. = 180 = 12h., Decl. = 0.From Sept. 23 to Dec. 22 the Sun is south of the equator,and its south declination is increasing.On December 22ud the Sun has described 270 from T,and is at L (Fig. 16). This is called the Winter Solstice.We have t L = 90, and the triangle . RL has two right

angles at R, L (Sph. Geom. 21). The Sun's polar dis-tance LP is a maximum (Sph. Geom. 26), and*R = L = 90, LR = / L^R = i. HenceSun's longitude = 270, R.A. = 270 = 18h.,

S. Decl. = i, (a maximum).From December 22 to March 21 the Sun's declination is

still south, but is decreasing.Finally, on March 21, when the Sun has performed a com-

plete circuit of the ecliptic, we have .Sun's long. = 360, B.A. = 360 = 24h., Decl. = 0.The longitude and R.A. are again reckoned as zero, and

they, together with the declination, undergo the same cycleof changes in the following year.


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22 ASTEONOMT.31. Sun's Variable Motion in R.A. We observe that

the Sun's right ascension is equal to its longitude four timesin the year, viz., at the two equinoxes and the two solstices.At other times this is not the case.For example, between the vernal equinox and summersolstice we have T-3f< T$, .'. Sun's E.A. < longitude.

Hence, even if the Sun's motion in longitude be supposeduniform, its R.A. will not increase quite uniformly. Thereis a further cause of the want of uniformity, namely, thatthe Sun's motion in longitude is not quite uniform ; but thisneed not be considered in the present chapter.

32. Direct and Retrograde Motions. The directionof the Sun's annual revolution relative to the stars, i.e., motionfrom west through south to east, is called direct. Theopposite direction, that of the diurnal apparent motions of thestars or revolution from east to west, is called retrograde.The revolutions of all bodies forming the solar system,with the exception of some comets and one or two smallsatellites, are direct.We shall see in Chapter III. that the apparent retrogradediurnal motion may be accounted for by the direct rotationpf the Earth about its polar axis,


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THE CELESTIAL SPHERE. 2333. Equinoctial and Solstitial Points Colures.From 30 it appears that the Summer andWinter Solstices

may be defined as the times of the year when the Sun attainsits greatest north and south declinations respectively. Thecorresponding positions of the Sun in the ecliptic ((7, Z,Fig. 17) are called the Solstitial Points. In the same waythe Equinoctial Points (T, ) are the positions of theSun at the Vernal and Autumnal Equinoxes when itsdeclination is zero.The declination circle PTP'^j passing through the equi-

noctial points, is called the Equinoctial Colure. Thedeclination circle PCP'L, passing through the solstitial points,is called the Solstitial Colure. The latter passes throughthe poles of the ecliptic (7T, K').

34. To find the Sun's Right Ascension and Decli-nation. In the "Nautical Almanack,"* the Sun's R.A.and declination at noon are tabulated for every day of theyear. Their hourly variations are also given in an adjoiningcolumn. To find their values at any time of the day,we only have to multiply the hourly variation by thenumber of hours that have elapsed since the preceding noon,and add to the value at that noon.EXAMPLE. Tfl find the Sun's R.A. and decl. on September 4, 1891

at 5h. 18m. in^gjs^ afternoon. We find from the Almanack for 1891under Septembers :Sun's R.A. a*oon = lOli. 52m. 15s., hourly variation 9'04s.N. Decl. at noon = 7 12' 12" 55'4"(1) RA. at noon = lOh. 52m. 15s.Increase in 5h. = 9'04s. x 5 = 45*2

18m. = 27.-. R.A. at 5h. 18m. - lOh. 53m. 3s.

(2) From the Almanack, decl. is less on September 5, and istherefore decreasing.N. Decl at noon = 7 12' 12"Decrease in 6h. = 55'4" x 5 = 4' 37" \ To be18m. - 17") subtracted.

N. Decl. at 6h. 18m. = 7 T 18 '* Also in " Whitaker's Almanack," which may be consulted with

advantage.


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24 ASTRONOMY.35. Rough Determination of the Sun's R.A. "We

can, without the "Nautical Almanack," find to within adegree or two, the Sun's E.A. on any given date, as follow^ :A year contains 365 days. In this period the Sun's E.A.increases by 360. Hence its average rate of increase is verynearly 30 per month, or 1 per day.Knowing the Sun's E.A. at the nearest equinox or solstice,we add 1 for every day later, or subtract 1 for every daybefore that epoch. If the E.A. is required in time, we allowfor the increase at the rate of 2h. per month, or 4m. per day.EXAMPLES. 1. To find the Sun's R.A. on January 1st. OnDecember 22nd the R.A. = 18h. Hence on January 1st, which isten days later, the Sun's R.A. = 18h. 40m.2. To find on what date the Sun's R.A. is lOh. 36m. On Sep-tember 23rd the R.A. is 12h. Also 12h.-10h. 36m. = 84m., andthe R.A. increases Sim. in 21 days. Hence the required date is 21

days before September 23, i.e., September 2nd,36. Solar Time. Apparent Noon is the time of the

Sun's upper transit across the meridian, that is, in northlatitudes, the time when the Sun souths. Apparent Mid-night is the time of the Sun's transit across the meridianbelow the pole (and usually below the horizon).An Apparent Solar Day is the interval between twoconsecutive apparent noons, or two consecutive midnights.Like the sidereal day, the solar day is divided into 24 hours,which are again divided into 60 minutes of 60 seconds each.For ordinary purposes the day is divided into two portions :the morning, lasting from midnight to noon ; the evening,from noon till midnight ; and in each portion times arereckoned from Oh. (usually called 12h.) up to 12h. Forastronomical purposes we shall find it more convenient tomeasure the solar time by the number of solar hours thathave elapsed since the preceding noon. Thus, 6.30 A.M. onJanuary 2nd will be reckoned, astronomically, as 18h. 30m.on January 1st. On the other hand, 12.53 P.M. will bereckoned as Oh. 53m., being 53 minutes past noon.During a solar day the Sun's hour angle increases from

to 360. It therefore increases at the rate of 15 per hour.HenceThe apparent solar time = the Sun's hour angleexpressed in time.


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THE CELESTIAL SPHERE. 25At noon the Sun is on the meridian. The sidereal time,

being the hour angle of T, is the same as the Sun's H.A., i.e.,Sidereal time of apparent noon Sun's R.A. at noon.At any other time, the difference between the sidereal andsolar times, being the difference between the hour angles ofT and the Sun, is equal to the Sun's E.A. Hence, as in25, we have

(Sidereal time) (apparent solar time) = Sun's R.A.If a and a+ x are the right ascensions of the Sun at two

consecutive noons, then, since a whole day has elapsed betweenthe transits, the total sidereal interval is 24h. +#, and exceeds asidereal day by the amount x. But the interval is a solar day.

Hence, the solar day is longer than the siderealday, and the difference is equal to the sun's dailymotion in R.A.*

37. Morning and Evening Stars. Sunrise andSunset. "When a star rises shortly before the Sun, and inthe same part of the horizon, it is called a Morning Star.Such a star is then only visible for a short time before sunrise.When a star sets shortly after the Sun, and in the same partof the horizon, it is called an Evening Star. It is thenonly visible just after sunset.It will be readily seen from a figure, that a star will be amorning star if its decl. is nearly the same as the Sun's, whileits E/.A. is rather less. Similarly, a star will be an eveningstar if its decl. is nearly the same as the Sun's, but its RA.somewhat greater. Thus, as the Sun's R.A. increases, thestars which are evening stars will become too near the Sun toto be visible, and will subsequently reappear as morning stars.The times of sunrise and sunset are calculated in themanner described in 29. The hour angles of the Sun, whencrossing the eastern and western horizons, determine theintervals of solar time between sunrise, apparent noon, andsunset. The two intervals are equal, if the Sun's decl. besupposed constant from sunrise to sunset a result veryapproximately true, since the change of decl. is always verysmall.

* Owing to the sun's variable motion in R. A., the apparent solar day is not quiteof constant length. In the present chapter, however, it may be regarded asapproximately constant.


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26 ASTRONOMY.38. The Gnomon. Determination of Obliquity of

Ecliptic. The Greek astronomers observed the Sun'smotion by means of the Gnomon, an instrument consistingessentially of a vertical rod standing in the centre of a hori-zontal floor. The direction of the shadow cast by the Sundetermined the Sun's azimuth, while the length of the shadow,divided by the height of the rod, gave the tangent of theSun's zenith distance. To find the meridian line, a circle wasdescribed about the rod as centre, and the directions of theshadow were noted when its extremity just touched the circlebefore and after noon. The sun's Z.D.'s at these twoinstants being equal, their azimuths were evidently (Sph.Geom. 27) equal and opposite, and the bisector of the anglebetween the two directions was therefore the meridian line.The Sun's meridian zenith distances were then observedboth at the summer solstice, when the Sun's IS", decl. is i andmeridian Z.D. least, and at the winter solstice, when the Sun'sS. decl. is i and meridian Z.D. greatest. Let these Z.D.'s be z land s2 respectively, and let I be the latitude of the place ofobservation. From 24, we readily see that

2 t = l-i, 22 = Z+t,/: *=*(.+*,), * = i(v-i);.thus determining both the latitude and the obliquity.

39. The Zodiac. The position of the ecliptic was definedby the ancients by means of the constellations of the Zodiac,which are twelve groups of stars, distributed at about equaldistances round a belt or zone, and extending about 8 oneach side of the ecliptic. The Sun and planets were observedto remain always within this belt. The vernal and autumnalequinoctial points were formerly situated in the constellationsof Aries and Libra, whence they were called the First Pointof Aries and the First Point of Libra. Their positions are veryslowly varying, but the old names are still retained. Thus,the " First Point of Aries" is now situated in the constel-lation Pisces.The early astronomers probably determined the Sun'sannual path by observing the morning and evening stars.After a year the same morning and evening stars would be

observed, and it would be concluded that the Sun performeda complete revolution in the year.


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THE CELESTIAL SPHEEE. 2740. Motion of the Moon. The Moon describes among

the stars a great circle of the celestial sphere, inclined tothe ecliptic at an angle of about 5. The motion is direct,and the period of a complete " sidereal " revolution is about27 days.In this time the Moon's celestial longitude increases by 360."When the Moon has the same longitude as the Sun, it issaid to be New Moon, and the period between consecutivenew Moons is called a Lunation. AVhen the Moon hasdescribed 360 from new Moon, it will again be at the samepoint among the stars ; but the Sun will have moved forward,so that the Moon will have a little further to go before itcatches up the Sun again. Hence the lunation will be ratherlonger than the period of a sidereal revolution, being about29\ days.The Age of the Moon is the number of days which haveelapsed since the preceding new Moon. Since the Moonseparates 360 from the Sun in 29j days, it will separate atthe rate of about 12, or more accurately 12-|- , per day,or 30' per hour. This enables us to calculate roughly theMoon's angular distance from the Sun, when the age of theMoon is given, and conversely, to determine the Moon's agewhen its angular distance is given.

EXAMPLE. On September 23, 1891, the Moon is 20 days old.To find roughly its angular distance from the Sun and its longitudeon that day.(1) In one day the Moon separates 12^- from the Sun; therefore,in 20 days it will have separated 20 x 121, or 244, and this is therequired angular distance from the Sun.

(2) On September 23 the Sun's longitude is 180 ; therefore theMoon's longitude is 180 + 244 = 424 = 360 + 64, or 64.This method only gives very rough results; for the Moon'smotion is far from uniform, and the variations seem very

irregular.Moreover, the plane of the Moon's orbit is not fixed, butits intersections with the ecliptic (called the Nodes) have aretrograde motion of 19 per year. Hence, for rough pur-poses, it is better to neglect the small inclination of the Moon'sorbit, and to consider the Moon in the ecliptic. If greateraccuracy be required, the Moon's decl. and R.A. may befound from the Nautical Almanack.


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28 ASTRONOMY.41. Astronomical Diagrams and Practical Applica-tions. We can now solve many problems connected withthe motion of the celestial bodies, such as determining the direc-tion in which a given star will be seen from a given place, at

a given time, on a given date, or finding the time of day atwhich a given star souths at a given time of year."We have, on the celestial sphere, certain circles, such asthe meridian, horizon, and prime vertical, also certain points,such as the zenith and cardinal points, whose positions relativeto terrestrial objects always remain the same. Besides these,we have the poles and equator, which remain fixed, withreference loth to terrestrial objects and to the fixed stars."We have also certain points, such as the equinoctial points,and certain circles, such as the ecliptic, which partake ofthe diurnal motion of the stars, performing a retrograderevolution about the pole once in a sidereal day. Lastly,we have the Sun, which moves in the ecliptic, performingone retrograde revolution relative to the meridian in a solarday, or one direct revolution relative to the stars in a year,and whose hour angle measures solar time.

In drawing a diagram of the celestial sphere, the positionsof the meridian, horizon, zenith, and cardinal points shouldfirst be represented, usually in the positions shown in Pig.18. Knowing the latitude nP of the place, we find thepole P. The points Q, ft, where the equator cuts the meri-dian, are found by making PQ = PR = 90 ; and the pointsQ, Ii, with E, W, enable us to draw the equator.We now have to find the equinoctial points. How to dothis depends on the data of the problem. Thus we mayhave given

(i.) The sidereal time ;(ii.) The hour angle of a star of known E.A. and decl ;

(iii.) The time of (solar) day and time of year.In case (i.), the sidereal time multiplied by 15 gives, in

degrees, the hour angle (Qf) of the first point of Aries.Measuring this angle from the meridian westwards, we findAries, and take Libra opposite to it. Any star of knowndecl. and R.A. can be now found by taking on the equator= star's R.A., and taking on MP, MX = star's decl.


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THE CELESTIAL SPHERE. 29The ecliptic may be drawn passing through Aries and

Libra, and inclined to the equator at an angle of about 23\(just over right angle). As we go round from west to east, orin the direct sense, the ecliptic passes from south to north ofthe equator at Aries ; this shows on which side to representthe ecliptic. Knowing the time of year, we now find theSun (roughly) by supposing it to travel to or from thenearest equinox or solstice about 1 per day from west to east.Finally, if the Moon's age be given, we find the Moon bymeasuring 12-i- per day, or 30' per hour eastwards from theSun.

P'

FIG. 18.In case (ii.), we either know the hour angle, QMoi QPMof.a known star (#), or, what is the same thing, the siderealinterval since its transit ; or, in particular, it is given that the

star is on the meridian. Each of these data determines J/~,the foot of the star's declination circle. FromMwe measurewestwards equal to the star's R.A. This finds Aries.


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80 ASTRONOMY.fn case (iii-)> the solar time multiplied by 15 gives the-Hun's hour angle QPS in degrees. From the time of year

we can find the Sun's R.A., TJPS. From these we findQ,PT and obtain the position of Aries just as in case (ii.)It will be convenient to remember that azimuth and hour

angle are measured from the meridian westwards, whileright ascension and celestial longitude are measured from thefirst point of Aries eastwards. Thus, since the Sun's diurnalmotion is retrograde, and its annual motion direct, the Sun'sazimuth, hour angle, R.A., and longitude are all increasing.Most problems of this class depend for their solution chieflyon the consideration of arcs measured along the equator, or(what amounts to the same) angles measured at the pole.In another class of problems depending on the relation be-tween the latitude, a star's decl. and meridian altitude ( 24),we have to deal with arcs measured along the meridian.These two classes include nearly all problems on the celestialsphere which do not require spherical trigonometry.

EXAMPLES.1. To represent, in a diagram, the positions of the Sun and Moon,and the star Herculis as seen by an observer in London on Aug. 19,

1891, at 8 p.m., the following data being given : Latitude of London-= 51, Moon's age at noon on Aug. 19 = 14 days 19 hours, Moon'slatitude = 2 S., K.A. of (Herculia = 16h. 37m., decl. = 31 48' N.The construction must be performed in the following order :(i.) Draw the observer's celestial sphere, putting in the meridian,

horizon, zenith Z, and four cardinal points n, E, s, W.(ii.) Indicate the position of the pole and equator. The observer' s-latitude is 51. Make, therefore, nP = 51. P will be the pole. TakePQ = PR = 90, and thus draw the equator, QERW.(Hi.) Find the declination circle passing through the Sun. The-time of day is 8 p.m. Therefore the Sun's hour angle is 8 x 15, or120. On the equator measure QK = 120 westwards from the-meridian. Then the Sun Q will lie on the declination circle PK.Since QW = 90, we may find K by taking WK = 30 = $ WR.(iv.) Find the first points of Aries and Libra. The date of obser-vation is August 19. Now, on September 23 the Sun is at =2=. Also-from August 19 to September 23 is 1 month 4 days. In this-interval the Sun travels about 34 from west to east. Hence theSun is 34 west of rO=. And we must measure K* = 34 eastwards^from 8, and thus find z.The first point of Aries ( T ) is the opposite point on the equator..


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THE CELESTIAL SPHERE. 31(v.) We may now draw the ecliptic Cri^= passing through the

first points of Aries and Libra, and inclined to the equator at anangle of about 23 (i.e., slightly over of a right angle). The Sunis above the equator on August 19; hence the ecliptic cuts PK aboveK. This shows on which side of the equator the ecliptic is to be-drawn ; we might otherwise settle this point by remembering thatthe ecliptic rises above the equator to the east of T .The intersection of the ecliptic with PE determines Q, the positionof the Sun.

FIG. 19.

ascenfion is 16h. 37m., in time, = 249 15' in angular measure. Onthe equator measure off TM = 249 15' in the direction west to east(i.e., the direction of direct motion) from T ; we must, therefore,take ^=M = 69 15'. On the declination circle HP, measure offMX = 31 48' towards P. Then x is the required position ofHerculis.

(vii.) Find the Moon. At 8 p.m. the Moon's age is 14d. 19h + 8h.= 15d. 3h. Hence, the Moon has separate/! from the Sun byabout 185 in the direction west to east. Measure off }) = 185from west to east, and put in }) about 2 below the ecliptic. TheMoon's position is thus found.


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32 ASTRONOMY.

a/-

2. To find (roughly) at what time of year the Star o Cygni(R.A. = 20h. 38m., clecl. = 44 53' N.) souths at 7 p.m.

Let o be the position of the star on the meridian (Fig 20). At7 p.m. the Sun's western hour angle (QS or QPS) = 7h. = 105.Also TEQ, the Star's R.A. = 20h.38m. Hence rRS, the Sun's R.A.= 20h. 38m. - 7h. = 13h. 38m. ; or,in angular measure, Sun's R.A.= 204 30'. Now, on September 23,Sun's R.A. = 180, and it increases atabout 1 per day. Hence the Sun'sR.A. will be 204 about 24 days later,i.e., about October 17th.

3. At noon on the longest day (June21) a vertical rod casts on a horizontalplane a shadow whose length is equal pIG 20to the height of the rod. To findthe latitude of the place and the Sun's altitude at midnight.

FIG. 21.

From the data, the Sun's Z.D. at noon, Z, evidently = 45.Also, if QR be the equator, 0Q = Sun's decl. = i = 23 27' (approx.);

.-. latitude of place = ZQ = 45 + 23 27' = 68 27'.If ' be the Sun's position at midnight,

P0' = PQ = 90-2.327' = G6 33'.But Pn = lat. = 68 27'.

... Q'w = 68 27'-66 33' = 1 54';and the Sun will be above the horizon at an alt. of 1 54' atmidnight.


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THE CELESTIAL SPHERE.

EXAMPLES. I.1. Why are thefollowing definitions alone insufficient? Tlie zenithand nadir are the poles of the horizon. The horizon is the greatcircle of the celestial sphere whose plane is perpendicular to the

line joining the zenith and nadir.2. The R.A. of an equatorial star is 270 ; determine approximatelythe times at which this star rises and sets on the 21st June. Inwhat quarter of the heavens should we look for the star at mid-

night ?3. Explain how to determine the position of the ecliptic relatively -to an observer at a given hour on a given day. Indicate the position .of the ecliptic relatively to an observer at Cambridge at 10 p.m. atthe autumnal equinox. (Lat. of Cambridge = 52 12' 51'6".) VV!

i4. Prove geometrically that the least of the angles subtended atan observer by a given star and different points of the horizonthat which measures the star's altitude.5. Show that in latitude 52 13' N. no circumpolar star when

southing can be within 75 34' of the horizon.C. Represent in a figure the position of the ecliptic at sunrise onMarch 21st as seen by an observer in latitude 45. Also in lati-tude 67. ,7. If the ecliptic were visible in the first part of the preceding

question, describe the variations which would take place during theday in the positions of its points of intersection with the horizon.

8. Determine when the star whose declination is 30" N. and whose .E.A. is 356 will cross the meridian at midnight.

9. The declination and R.A. of a given star are 22 N. and6h. 20m. respectively. At what period of the year will it be (i.) amorning, (ii.) an evening star ? In what part of the sky would youthen look for it ?

10. Find the Sun's R.A. (roughly) on January 25th, and thus de-termine about whatxtime Aldebaran (R.A. 4h. 29m.) will cross themeridian that night.11. Where and at what time of the year would you look forFomalhaut ? (R.A. 22h. 51m., decl. 30. 16' S.)12. At the summer solstice the meridian altitude of the Sun is75. What is the latitude of the place ? What will be the meridian

altitude of the Sun at the equinoxes and at the winter solstice ?~


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34 ASTRONOMY.

EXAMINATION PAPER. I.1. Explain how the directions of stars can be represented bymeans of points on a sphere. Explain why the configurations ofthe constellations do not depend on the position of the observer,and why the angular distance of two different bodies on the celestial

sphere gives no idea of the actual distance between them.2. Define the terms horizon, meridian, zenith, nadir, equator,

ecliptic, vertical, prime vertical, and represent their positions in afigure.

3. Explain the use of coordinates in fixing the position of a bodyon the celestial sphere, and define the terms altitude, azimuthtpolar distance, hour angle, right ascension, declination, longitude,latitude. Which of these coordinates alwa3Ts remain constant forthe same star ?

4. Define the obliquity of the ecliptic and the latitude of theobserver. Give (roughly) the value of the obliquity, and of the latitudeof London. Indicate in a diagram of the celestial sphere twelvedifferent arcs and angles which are equal to the latitude of theobserver.

5. What is meant by a sidereal day and a sidereal hour ? Howcould you find the length of a sidereal day without using a tele-scope ? Why is sidereal time of such great use in connection withastronomical observations ?

6. Show that the declination and right ascension of a celestialbody can be determined by meridian observations alone.

7. What is meant by a circumpolar star ? What is the limit ofdeclination for stars which are circumpolar in latitude 60 N. ?Indicate in a diagram the belt of the celestial sphere containing allthe stars which rise and set.8. Define the terms year, equinoxes, solstices, equinoctial and

solstitial points, equinoctial and solstitial colures. What are thedates of the equinoxes and solstices, and what are the correspondingvalues of the Sun's declination, longitude, and right ascension?Find the Sun's greatest and least meridian altitudes at London.9. Why is it that the interval between two transits of the Sun orMoon is rather greater than a sidereal day ? Show how the Sun'sR.A. may be found (roughly) on any given date, and find it on

July 2nd, expressed in hours, minutes, and seconds.10. Indicate (roughly) in a diagram the positions of the following

stars as seen in latitude 51 on July 2nd at 10 p.m, : Capella (R.A.5h. 8m. 38s., decl. 45 53' 10" N.), a Lyras (R.A. 18h. 33m. 14s.,decl. 38 40' 57" N.), a Scorpii (R.A. 16h. 22m. 43s., decl. 26 11'22" S.), a Ursse Majoris (R.A. lOh. 57m. Os., dec!. 62 20' 22" N.)


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CHAPTER II.THE OBSERYATOHY.

SECTION I. Instruments adaptedfor Meridian Observations.42. One of the most important problems of practical astro-

nomy is to determine, by observation, the right ascension anddeclination of a celestial body. We have seen in Chapter I.that these coordinates not only suffice to fix the position of astar relative to neighbouring stars, but they also enable us tofind the direction in which the star may be seen from a givenplace at a given time of day on a given date (41). More-over, it is evident that by determining every day the decli-nation and right ascension of the Sun, the Moon, or a planet,the paths of these bodies relative to the stars can be mappedout on the celestial sphere and their motions investigated.In Section II. of the preceding chapter we showed thatthe right ascension and declination of a star can be deter-mined by observations made when the star is on the meridian.We proved the following results :The star's R.A. measured in time is equal to the time oftransit indicated by a sidereal clock ( 24).The star's north decl. d can be found from z its meridianzenith distance, and I the latitude of the observatory by theiormula d = l+z,where if the decl. is south d is negative, and if the star tran-sits south of the zenith z is negative (24).

Lastly, I can be found by observing the altitudes of acircumpolar star at its two culminations, and is thereforeknown ( 28).Hence the most essential requisites of an observatory mustinclude (i.) a clock to measure sidereal time, (ii.) a telescopeso fitted as to be always pointed in the meridian, providedwith graduated circles to measure its inclination to the ver-tical, and with certain marks to fix the position of a star inits field of view.


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36 ASTRONOMY.43. The Astronomical Clock is a clock regulated to

indicate sidereal time. It should be set to mark Oh. Om. Os.at the time when the first point of Aries crosses the meridian.It will therefore gain about 4 minutes per dayon an ordinary clock, or a whole day in thecourse of a year ( 22, 36).The clock is provided with a seconds hand, andthe pendulum beats once every second, produc-ing audible "ticks"; hence an observer canestimate times by counting the ticks, whilst heis watching a star through a telescope.The pendulum is a compensating pendu-lum, or one whose period of oscillation is un-affected by changes of temperature. The formmost commonly used is Graham's MercurialPendulum, in which the bob carries two glasscylinders containing mercury (Fig. 22). If thetemperature be raised, the effect of. the increasein length of the pendulum rod is compensatedfor by the mercury expanding and rising in thecylinders. The same result is also effected inHarrison's Gridiron Pendulum, described inWallace Stewart's Text-Boole of Heat, page 37.The clock is sometimes regulated by placingsmall shot in a cup attached to the pendulum.

FIG. 23.


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THE OBSERVATORY. 3744. The Astronomical Telescope (Fig. 23) consists

essentially of two convex lenses, or systems of lenses, and0', fixed at opposite ends of a metal tube, and called theobject-glass and eye-piece respectively. The former lensreceives the rays of light from the stars or other distant objects,and forms an inverted " image " (al) of the objects. Thecentre of the round object-glass is. called its " opticalcentre," and the image is produced as follows: Let AAAbe a pencil of rays from a distant star. By traversing theobject-glass these rays are refracted or bent towards themiddle ray A 0, which alone is unchanged in direction. Therays all converge to a common point or "focus'' at a point ain A produced, and, if received by the eye after passing #,they would appear to emanate from a luminous point or" image " of the star at a.

Similarly, the rays BBB, coming from another distant star,will converge to a focus at a point b in BO produced, andwill give the effect of an

" image" of the star at b. Allthese images (a, b) lie in a certain plane FN, called the focalplane of the object-glass, and they form a kind of picture orimage of such stars as are in the field of view.The eye-piece 0' acts as a kind of magnifying glass, andenlarges the image ab just as if it were a small object placedin the focal plane FN. The figure shows how a second imageA'B' is formed by the direction of the pencils of light afterrefraction through (/. This is the final image seen on lookingthrough the telescope. The eye must be placed in the planeEE, so as to receive the pencils from A', B'.

If, now, a framework of fine wires or spider's threads(Fig. 25) be stretched across the tube in the focal planeFNj these wires, together with the image (#J), will beequally magnified by the eye-piece. They will thus beseen in focus simultaneously with the stars, and the fieldof view will appear crossed by a series of perfectly distinctlines, which will enable us to fix any star's position, andthus determine its exact direction in space. Suppose, forexample, that we have two wires crossing one another at thepoint F', and the telescope is so adjusted that the image of astar coincides with F', then we know that the star lies in theline joining F' to the optical centre of the object-glass.


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00 ASTRONOMY.45. The Transit Circle (Figs. 24, 26) is the instrumentused for determining both right ascension and declination. It

consists of a telescope, ST, attached perpendicularly to alight, rigid axis, WPPE, hollow in the interior. The ex-tremities of this axis are made in the form of cylindrical pivots,E, W, which are capable of revolving freely in two fixed forks,called Y's, from their shape. These Y's rest on piers of solidstone, built on the firmest possible foundations, and they arecarefully fixed, so as always to keep the axis exactly hori-zontal and pointing due east and west.

FIG. 24.In order to dimini?0i the effect of friction in wearing awaythe pivots, the axis is also partially supported at P, P upon

friction rollers (not represented in the figure) attached to a


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THE OBSERVATORY. 3system of levers ( Q, Q) and counterpoises (R, R) placed withinthe piers. These support about four-fifths of the weight ofthe telescope, leaving sufficient pressure on the Y's to ensure-their keeping the axis fixed.Within the telescope tube, in the focal plane of the object-glass ( 44), is fixed a framework of cross wires, presenting^the appearance shown in Fig. 25. Five, or sometimes seven,wires appear vertical, and two appear horizontal. Of thelatter, one bisects the field of view ; the other is movable upand down by means of a screw, whose head is divided bygraduation marks which indicate the position of the wire.The line joining the optical centre of the object-glass tothe point of intersection of the middle vertical wire with the-fixed horizontal wire is called the Line ofColliinatiou. The wires should be soadjusted that the line of colliination is per-pendicular to the axis about which thetelescope turns. For this purpose theframework carrying the wires can be movedhorizontally, by means of a screw, into theright position. If the Y's have been accu-rately fixed, then, as the telescope turns,the line of collimation will always lie in the plane of themeridian. Hence, when a star transits we shall, on lookingthrough the telescope, see it pass across the middle vertical,wire.

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Elementary Mathematical Astronomy, Barlow & Bryan